GENERALIZED FOUR-NODE PLANE RECTANGULAR AND QUADRILATERAL ELEMENTS AND THEIR APPLICATIONS IN THE MULTISCALE ANALYSIS OF HETEROGENEOUS STRUCTURES

Homogenization methods have been widely used in the multiscale analysis of heterogeneous structures. For two-dimensional problems, microscale and macroscale computations in these methods are mainly conducted based on conventional plane rectangular and isoparametric elements due to their simplicity. However, since the coupled deformation among different directions is intrinsically ignored in these elements, they may reduce the efficiency of the homogenization methods and thus a dense mesh needs to be used to obtain more reliable and accurate results. To overcome these deficiencies, generalized rectangular and quadrilateral elements with four nodes for plane problems are developed in this paper. The coupled additional terms are introduced in the interpolation shape functions without increasing the number of degrees of freedom of the elements. Based on the elastic equations of equilibrium within the elements, analytical formulas of these functions are derived under linear boundary conditions. It is demonstrated that two kinds of elements can represent three rigid body modes and ensure the passage of the patch test for the requirement of convergence; and are all compatible. In addition, several elements with different forms of the coupled additional terms are also constructed. The verification and accuracy of the new developed elements are examined by means of numerical examples. The homogenization analysis for two-dimensional heterogeneous structures based on the developed elements is performed and the advantages of these elements over the conventional four-node plane rectangular and isoparametric quadrilateral elements are discussed. It is demonstrated that the new elements can be successfully used for the multiscale analysis of heterogeneous structures.